81 research outputs found
Finding Fair and Efficient Allocations
We study the problem of allocating a set of indivisible goods among a set of
agents in a fair and efficient manner. An allocation is said to be fair if it
is envy-free up to one good (EF1), which means that each agent prefers its own
bundle over the bundle of any other agent up to the removal of one good. In
addition, an allocation is deemed efficient if it satisfies Pareto optimality
(PO). While each of these well-studied properties is easy to achieve
separately, achieving them together is far from obvious. Recently, Caragiannis
et al. (2016) established the surprising result that when agents have additive
valuations for the goods, there always exists an allocation that simultaneously
satisfies these two seemingly incompatible properties. Specifically, they
showed that an allocation that maximizes the Nash social welfare (NSW)
objective is both EF1 and PO. However, the problem of maximizing NSW is
NP-hard. As a result, this approach does not provide an efficient algorithm for
finding a fair and efficient allocation.
In this paper, we bypass this barrier, and develop a pseudopolynomial time
algorithm for finding allocations that are EF1 and PO; in particular, when the
valuations are bounded, our algorithm finds such an allocation in polynomial
time. Furthermore, we establish a stronger existence result compared to
Caragiannis et al. (2016): For additive valuations, there always exists an
allocation that is EF1 and fractionally PO.
Another contribution of our work is to show that our algorithm provides a
polynomial-time 1.45-approximation to the NSW objective. This improves upon the
best known approximation ratio for this problem (namely, the 2-approximation
algorithm of Cole et al. (2017)). Unlike many of the existing approaches, our
algorithm is completely combinatorial.Comment: 40 pages. Updated versio
EFX Exists for Three Agents
We study the problem of distributing a set of indivisible items among agents
with additive valuations in a manner. The fairness notion under
consideration is Envy-freeness up to any item (EFX). Despite significant
efforts by many researchers for several years, the existence of EFX allocations
has not been settled beyond the simple case of two agents. In this paper, we
show constructively that an EFX allocation always exists for three agents.
Furthermore, we falsify the conjecture by Caragiannis et al. by showing an
instance with three agents for which there is a partial EFX allocation (some
items are not allocated) with higher Nash welfare than that of any complete EFX
allocation.Comment: Full version of a paper published at Economics and Computation (EC)
202
Fairness in Power Flow Network Congestion Management with Outer Matching and Principal Notions of Fair Division
The problem of network flow congestion occurring in power networks is increasing in severity. Especially in low-voltage networks this is a novel development. The congestion is caused for a large part by distributed and renewable energy sources introducing a complex blend of prosumers to the network. Since congestion management solutions may require individual prosumers to alter their prosumption, the concept of fairness has become a crucial topic of attention. This paper presents a concept of fairness for low-voltage networks that prioritizes local, outer matching and allocates grid access through fair division of available capacity. Specifically, this paper discusses three distinct principal notions of fair division; proportional, egalitarian, and nondiscriminatory division. In addition, this paper devises an efficient algorithmic mechanism that computes such fair allocations in limited computational time, and proves that only egalitarian division results in incentive compatibility of the mechanism
Consensus-Halving: Does It Ever Get Easier?
In the -Consensus-Halving problem, a fundamental problem in fair division, there are agents with valuations over the interval , and the goal is to divide the interval into pieces and assign a label "" or "" to each piece, such that every agent values the total amount of "" and the total amount of "" almost equally. The problem was recently proven by Filos-Ratsikas and Goldberg [2019] to be the first "natural" complete problem for the computational class PPA, answering a decade-old open question. In this paper, we examine the extent to which the problem becomes easy to solve, if one restricts the class of valuation functions. To this end, we provide the following contributions. First, we obtain a strengthening of the PPA-hardness result of [Filos-Ratsikas and Goldberg, 2019], to the case when agents have piecewise uniform valuations with only two blocks. We obtain this result via a new reduction, which is in fact conceptually much simpler than the corresponding one in [Filos-Ratsikas and Goldberg, 2019]. Then, we consider the case of single-block (uniform) valuations and provide a parameterized polynomial time algorithm for solving -Consensus-Halving for any , as well as a polynomial-time algorithm for ; these are the first algorithmic results for the problem. Finally, an important application of our new techniques is the first hardness result for a generalization of Consensus-Halving, the Consensus--Division problem. In particular, we prove that -Consensus--Division is PPAD-hard
Petition for Paternity and Maintenance (Note de lege ferenda)
Digitalizacja i deponowanie archiwalnych zeszytów RPEiS sfinansowane przez MNiSW w ramach realizacji umowy nr 541/P-DUN/201
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